Material and Energy Balances: cram pack

Formula sheets, the key method steps for every topic and 8 practice problems, with full solutions at the back.

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1Formulas and methods

Mass balance

Steady state
in = out
Total balance
ṁ1 + ṁ2 = ṁ3
Component balance
ṁ1x1 + ṁ2x2 = ṁ3x3
Evaporator
F·xF = P·xP, W = F − P

Energy balance

Sensible heat
Q = ṁ·cp·ΔT
Latent heat
Q = ṁ·hfg
Mixing
ΣṁcpTin = ΣṁcpTout
Water
cp ≈ 4.18 kJ/(kg·K)

Ideal gas law

Ideal gas law
pV = nRT
Gas constant
R = 8.314 J/(mol·K)
Combined gas law
p1V1/T1 = p2V2/T2
Moles
n = m/M
Temperature
T[K] = T[°C] + 273.15

Stoichiometry

Moles
n = m/M
Concentration
c = n/V
Mole ratio
use the coefficients of the balanced equation
Limiting reactant
the one that runs out first
Percent yield
actual / theoretical × 100%

Method

  1. Write and balance the chemical equation.
  2. Convert the given mass to moles: n = m/M.
  3. Use the coefficients as the mole ratio to find the moles of the substance you want.
  4. Convert back to mass with m = n·M. With two reactants, find the limiting reactant first.

2Practice problems

1.Mass balanceMedium
MixerStream 1750 kg/h · 30% NaOHStream 2ṁ₂ = ? · 15% NaOHProduct20% NaOH

How much of stream 2 must be added each hour to stream 1 so that the blended product contains 20% NaOH? Find ṁ2.

ṁ1 = 750 kg/hx1 = 30 %x2 = 15 %x3 = 20 %
ṁ2 = kg/h
2.Mass balanceHard
EvaporatorFeed F700 kg/h · 5% sugarProduct PP = ? · 65%Vapour W (pure water)0% sugar

An evaporator concentrates a sugar solution by boiling off pure water. Find the product flow P in kg/h.

F = 700 kg/hxF = 5 %xP = 65 %
P = kg/h
3.Energy balanceMedium
hot sidecold sideglycol mix · 150 °C80 °Cwater · 15 °Ct = ? °C

Hot glycol mix cools water in a counter-flow heat exchanger with no heat loss. Find the water outlet temperature tout.

ṁh = 1.5 kg/scp,h = 3.6 kJ/(kg·K)Th: 150 → 80 °Cṁc = 3 kg/scp,c = 4.18 kJ/(kg·K)tin = 15 °C
tout = °C
4.Energy balanceHard
hot sidecold sideglycol mix · 160 °C70 °Cwater · 20 °C55 °C · ṁ = ?5% lost

Hot glycol mix heats water in an exchanger that loses 5% of the heat released by the hot stream to the surroundings. Find the water flow rate ṁc needed.

ṁh = 2.5 kg/scp,h = 3.6 kJ/(kg·K)Th: 160 → 70 °Ct: 20 → 55 °Ccp,c = 4.18 kJ/(kg·K)loss = 5 %
ṁc = kg/s
5.Ideal gas lawMedium
490 g of O₂V = 20 LT = 10 °CP = ?

A tank of 20 L contains 490 g of O₂ at 10 °C. Treat it as an ideal gas and find the pressure P.

m = 490 gM = 32 g/molV = 20 LT = 10 °CR = 8.314 J/(mol·K)
P = kPa
6.Ideal gas lawHard
CO₂, M = 44.01 g/molP = 400 kPaT = 100 °Cρ = ?

Find the density ρ of CO₂ at 400 kPa and 100 °C, treating it as an ideal gas.

M = 44.01 g/molP = 400 kPaT = 100 °CR = 8.314 J/(mol·K)
ρ = kg/m³
7.StoichiometryMedium
m(X)n(X)n(Fe)m(Fe)÷ M(X)× 2/ν× M(Fe)

6 g of Fe2O3 and 48 g of CO react according to Fe2O3 + 3CO → 2Fe + 3CO2. What mass m of Fe can form?

M(Fe2O3) = 159.69M(CO) = 28.01M(Fe) = 55.85 g/mol
m = g
8.StoichiometryHard
m(H2​)n(H2​)n(H2​O)m(H2​O)÷ M(H2​)× 2/2× M(H2​O)

In the reaction 2H2 + O2 → 2H2O, 4 g of H2 reacts completely (the other reactants are in excess) and 25.03 g of H2O is collected. Find the percent yield Y.

M(H2) = 2.016 g/molM(H2O) = 18.02 g/molactual: 25.03 g
Y = %

3Solutions

1.Mass balance

Answer: ṁ2 = 1500 kg/h

Steady-state mass balance: in = out for the total and for each component. Blending problems combine both.

Balances

Total: ṁ3 = ṁ1 + ṁ2
NaOH: ṁ1x1 + ṁ2x2 = (ṁ1 + ṁ2)x3

Solve for ṁ2

ṁ2 = ṁ1(x1 − x3)/(x3 − x2) = 750·(30 − 20)/(20 − 15) = 1500 kg/h

Answer

ṁ2 = 1500 kg/h
2.Mass balance

Answer: P = 53.85 kg/h

Tie component: a component that passes through unchanged in one stream (here the solute) links the flows directly. Then the total balance gives the rest.

Solute balance

F·xF = P·xP → P = 700·5/65 = 53.85 kg/h

Answer

P = 53.85 kg/h
3.Energy balance

Answer: tout = 45.14 °C

Exchanger energy balance: with no losses, ṁhcp,h(Th,in − Th,out) = ṁccp,c(tout − tin).

Heat released by the hot stream

Q = 1.5·3.6·(150 − 80) = 378 kW

Heat picked up by the water

tout = tin + Q/(ṁccp,c) = 15 + 378/(3·4.18) = 45.14 °C

Answer

tout = 45.14 °C
4.Energy balance

Answer: ṁc = 5.26 kg/s

Energy balance with losses: heat released = heat absorbed + heat lost.

Heat released

Qh = 2.5·3.6·(160 − 70) = 810 kW

Heat that reaches the water

Qc = (1 − 0.05)·810 = 769.5 kW

Water flow rate

ṁc = Qc/(cp,c·Δt) = 769.5/(4.18·35) = 5.26 kg/s

Answer

ṁc = 5.26 kg/s
5.Ideal gas law

Answer: P = 1802 kPa

Ideal gas law: PV = nRT, with n = m/M.

Moles

n = m/M = 490/32 = 15.31 mol
T = 10 + 273.15 = 283.1 K

Pressure

P = nRT/V = 15.31·8.314·283.1/0.02 = 1802 kPa

Answer

P = 1802 kPa
6.Ideal gas law

Answer: ρ = 5.674 kg/m³

Ideal gas density: ρ = PM/(RT).

Rearrange the gas law

PV = (m/M)RT → ρ = m/V = PM/(RT)
T = 100 + 273.15 = 373.1 K

Density

ρ = 400·44.01/(8.314·373.1) = 5.674 kg/m³

Answer

ρ = 5.674 kg/m³
7.Stoichiometry

Answer: m = 4.197 g

Stoichiometry: a balanced equation counts particles, not grams. Convert mass to moles with n = m/M, use the coefficients of the equation as the mole ratio, then convert back with m = n·M. With two reactants, the limiting reactant is the one that runs out first and fixes how much product forms.

Moles of each reactant

n(Fe2O3) = 6/159.69 = 0.03757 mol
n(CO) = 48/28.01 = 1.714 mol

Limiting reactant

0.03757/1 = 0.03757, 1.714/3 = 0.5712 → Fe2O3

Mass of product

n(Fe) = 0.03757·2 = 0.07515 mol
m = 0.07515·55.85 = 4.197 g

Answer

m = 4.197 g
8.Stoichiometry

Answer: Y = 70.01 %

Stoichiometry: a balanced equation counts particles, not grams. Convert mass to moles with n = m/M, use the coefficients of the equation as the mole ratio, then convert back with m = n·M. With two reactants, the limiting reactant is the one that runs out first and fixes how much product forms.

Theoretical yield

n(H2) = 4/2.016 = 1.984 mol
m = 1.984·2/2·18.02 = 35.75 g

Percent yield

Y = 25.03/35.75·100 = 70.01%

Answer

Y = 70.01%

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